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Welfare
Consider an economy with 2 individual & the total amount of food = 200 units,
has to be allocated between the 2.
1
𝑢1 = √𝐹1
𝑢2 = 2 √𝐹2
a.
b.
c.
d.
e.
If food is divided equally then, what will be their utilities?
How should food be allocated to equalize their utilities
How should food be allocated to maximize the sum of their utilizes.
If ū2 = 5, then what is 𝑢1 ?
If social welfare function is 𝑤 = √𝑢1 𝑢2 then how would food be allocated.
⋕ If PPF is given, 𝑥12 + 2𝑥22 = 900
a) If 𝐶 𝑛 is always in the form 𝑥2 = 2𝑥1 then how much of 𝑥1 &2𝑥2 purchased.
⋕ If two people derive their utility from each other.
2⁄
a)
b)
c)
d)
1⁄
1⁄
1⁄
𝑆𝐴 = 𝑆𝐴 3 𝑆𝐵 3
𝑢𝐵 = 𝑆𝐴 3 𝑆𝐴 3 & 𝑆𝑆 + 𝑆𝐵 = 24
How would ‘A’ choose 𝑆𝐴 &𝑆𝐵 .
How would ‘B’ choose 𝑆𝐴 &𝑆𝐵 .
What are pareto optimal allocations
If a Rawlasian person divides it then, how would he divide it.
⋕ A mother has two children. A&B. she has $ 100 to give between both. How
would she divide the money if both all equally efficient.
a. 𝑤 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 (𝑢𝐴 , 𝑢𝐵 )
b. 𝑤 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 (𝑢𝐴 , 𝑢𝐵 )
−1 −1
c. 𝑤 = 𝑢 𝑢
𝐴
𝐵
d. 𝑤 = √𝑢𝐴 + √𝑢𝐵
⋕ What if A was twice as efficient as B.
⋕ A’s utility function is same as 𝐵1 which is 𝑢 = 𝑥 2 + 𝑦 2 for their own 𝐶 𝑛 .
Calculate fair allocation.
⋕ A’s utility function 𝑢𝐴 = 2𝑥𝐴 + 𝑦𝐴
𝑢𝐵 = 𝑥𝐵 + 2𝑦𝐵
Find fair allocations.
Total 𝑥 = 12 = 𝑦
⋕ Given A’s
𝑢𝐵 = 𝑥𝐵 𝑦𝐵
(8,18)
𝑢𝐴 = 𝑥𝐴 𝑦𝐴
&
(12,2)
find equilibrium
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The government thinks equilibrium & thinks that (10,10) for both is more
equitable. Is this allocation optimal & is it equitable.
⋕ A’s
𝑢𝐴 = 𝑥𝐴 + 2√𝑦𝐴
𝑢𝐵 = 𝑥𝐵 + 4√𝑦𝐵
a. find UPF.
b. If SWF is 𝑤 = 𝑢𝐴 + 𝑢𝐴 , find their allocations.
Total 𝑥 = 𝑦 = 20
⋕ A & B are two individual A’s endowment is (6,3). B is not known.
𝑢𝐴 = 𝑥𝐴 + √12𝑦𝐴
𝑢𝐵 = 𝑥𝐵 + 3𝑦𝐵
Assume that there is competitive equilibrium find P & A’s 𝐶 𝑛 bundle.
⋕ Find fair allocations for the following
i.
𝑢𝐴 = 2𝑥𝐴
𝑢𝐵 = 𝑥𝐵 + 𝑦𝐵
2
ii.
𝑢𝐴 = 𝑥𝐴
𝑢𝐵 = 𝑥𝐵 + 𝑦𝐵
𝑦𝐴
iii.
𝑢𝐴 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 (𝑥𝐴 , 2 )
𝑢𝐵 = 4𝑥𝐵 + 3𝑦𝐵
(4,8)
(6,12)
Also find comp. equilibrium
⋕
𝑢𝐴 = 𝑥𝐴 𝑦𝐴
(40,160)
a. Government thinks (80,
𝑢𝐵 = 𝑥𝐵 𝑦𝐵2
(240,120)
140
700
3
3
) (200,
) is more equitable. Show that this is
pareto optimal also.
b. For the above find P=?
c. Assume there is a lumpsum transfer from B to A.
Find new & old equilibrium.
⋕
𝑢𝐴 = 2(𝑥𝐴 . 𝑦𝐴 )1/4
𝑢𝐵 = 2(𝑥𝐵 . 𝑦𝐵 )1/4
𝑤 = 100,100
a. Find UPF
b. Find Rawlsian SWF & allocation
1⁄
Lets say 𝑈𝑃𝐹 = 𝑢𝐴 2 + 𝑢𝐵 = 100. If SWF is weighted sum of utilities, the
welfare maximizing allocation is (2500,50). Find SWF.
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